Move meshes of several objects without affecting relative position
I would suggest one of two methods.

Method 1


Move your 3D cursor to the center or your objects (that you want to move)
Set origin of all the objects to 3D cursor.
Now move the objects. (movements are based on origin. You can also set transform to "individual origins for a different effect.)


Note: Selecting all objects and making a group will make things easier. This way you can select the whole group easily by selecting one of the objects and pressing shift+G then selecting "group".

Method 2


Create an empty
parent those objects to the empty.
Move the empty to the location you want.


Note: put the empty in the center of the object before parenting.

Method 3

This method is for moving the vertices of different objects to the same place, when the origins are already in the desired location.


Select all the objects you would like to edit.
Snap 3D cursor to selected (shift+s)
Set origin to geometry. (ctrl+alt+shift+c or toolbar)
Snap "selection to Cursor (offset)". (this operation is based on the origins, which are now at the centre of the geometry [mesh].)